Learning sparse optimal rule fit by safe screening

In this paper, we consider linear prediction models in the form of a sparse linear combination of rules, where a rule is an indicator function defined over a hyperrectangle in the input space. Since the number of all possible rules generated from the training dataset becomes extremely large, it has been difficult to consider all of them when fitting a sparse model. In this paper, we propose Safe Optimal Rule Fit (SORF) as an approach to resolve this problem, which is formulated as a convex optimization problem with sparse regularization. The proposed SORF method utilizes the fact that the set of all possible rules can be represented as a tree. By extending a recently popularized convex optimization technique called safe screening, we develop a novel method for pruning the tree such that pruned nodes are guaranteed to be irrelevant to the prediction model. This approach allows us to efficiently learn a prediction model constructed from an exponentially large number of all possible rules. We demonstrate the usefulness of the proposed method by numerical experiments using several benchmark datasets.